GROUP ACTIONS ON SPHERES WITH RANK ONE ISOTROPY

IAN HAMBLETON AND ERGÜN YALÇIN

Abstract. Let G be a rank two finite group, and let ℋ denote the family of all rank one p-subgroups of G, for which rank_{p}(G) = 2. We show that a rank two finite group G which satisfies certain group-theoretic conditions admits a finite G-CW-complex X ≃ S^{n} with isotropy in ℋ, whose fixed sets are homotopy spheres.

1. Introduction

Let G be a finite group. The unit spheres S(V) in finite-dimensional orthogonal representations of G provide the basic examples of smooth G-actions on spheres. Moreover, character theory reveals intricate relations between the dimensions of the fixed sets S(V)^{H}, for subgroups H ≤ G, and the structure of the isotopy subgroups {G_{x} | x ∈ S(V)}. We are interested in understanding how these basic invariants are constrained for smooth non-linear finite group actions on spheres.

We say that G has rank k if it contains a subgroup isomorphic to (Z/p)^{k}, for some prime p, but no subgroup (Z/p)^{k+1}, for any prime p. In this paper, we use chain complex methods to study the following problem, as a step towards smooth actions.

Question. For which finite groups G, does there exist a finite G-CW-complex X ≃ S^{n} with all isotropy subgroups of rank one ?

The isotropy assumption implies that G must have rank ≤ 2, by P. A. Smith theory (see Corollary 7.3). Since every rank one finite group can act freely on a finite complex homotopy equivalent to a sphere (Swan [13]), we will restrict to groups of rank two.

There is another group theoretical necessary condition related to fusion properties of the Sylow subgroups. This condition involves the rank two finite group Qd(p) which is the group defined as the semidirect product

Qd(p) = (Z/p × Z/p) ⋊ SL_{2}(p)

with the obvious action of SL_{2}(p) on Z/p × Z/p. In his thesis, Ünlü [16, Theorem 3.3] showed that Qd(p) does not act on a finite CW-complex X ≃ S^{n} with rank 1 isotropy. This means that any rank two finite group which includes Qd(p) as a subgroup cannot admit such actions.

More generally, we say Qd(p) is p′-involved in G if there exists a subgroup K ≤ G, of order prime to p, such that N_{G}(K)/K contains a subgroup isomorphic to Qd(p). The argument given by Ünlü in [16, Theorem 3.3] can be extended easily to obtain the stronger necessary condition (see Proposition 6.3):

Date: March 23, 2013.

Research partially supported by NSERC Discovery Grant A4000. The second author is partially supported by TÜBİTAK-TBAG/110T712.

(⋇). Suppose that there exists a finite G-CW-complex X ≃ S^{n} with rank 1 isotropy. Then Qd(p) is not p′-involved in G, for any odd prime p.

In the other direction, finite groups which do not p′-involve Qd(p) have some interesting complex representations. Jackson [9, Theorem 47] proved that, in this case, each Sylow p-subgroup of G has a p-effective character which respects fusion in G. We use these characters to reduce the isotropy from p-subgroups to rank one p-subgroups.

Let ℱ be a family of subgroups of G closed under conjugation and taking subgroups. For constructing group actions on CW-complexes with isotropy in the family ℱ, a good algebraic approach is to consider projective chain complexes over the orbit category relative to the family ℱ (see [6]).

Let 𝒮_{G} denote the set of primes p such that rank_{p}(G) = 2. Let ℋ_{p} denote the family of all rank one p-subgroups H ≤ G, for p ∈ 𝒮_{G}, and let ℋ = ⋃ {H ∈ ℋ_{p} | p ∈ 𝒮_{G}}. Our main result is the following:

Theorem A. Let G be a rank two finite group satisfying the following two conditions: (i) G does not p′-involve Qd(p) for any odd prime p ∈ 𝒮_{G}; (ii) if 1 ≠ H ∈ ℋ_{p}, then rank_{q}(N_{G}(H)/H) ≤ 1 for every prime q ≠ p.

Then there exists a finite G-CW-complex X with isotropy in ℋ, such that X^{H} is a homotopy sphere for each H ∈ ℋ.

Our construction produces new non-linear G-CW-complex examples, for certain groups G which do not admit any orthogonal representations V with rank one isotropy on the unit sphere S(V) (see Example 7.4). In Section 7, we give the motivation for condition (ii) on the q-rank of the normalizer quotients. It is used in a crucial way (at the algebraic level) in the construction of our actions, but it is not, in general, a necessary condition for the existence of a finite G-CW-complex X ≃ S^{n} with rank 1 isotropy (see Example 7.5). Determining the full list of necessary conditions is still an open problem.

Theorem A is an extension of our earlier joint work with Semra Pamuk [6] where we have shown that the first non-linear example, the permutation group G = S_{5} of order 120, admits a finite G-CW-complex X ≃ S^{n} with rank one isotropy. Theorem A implies this earlier result since for G = S_{5}, the set 𝒮_{G} includes only the prime 2 and the second condition above holds since all Sylow p-subgroups of S_{5} for odd primes are cyclic. More generally, we have:

Corollary B. Let p be a fixed prime and G be a finite group such that rank_{p}(G) = 2, and rank_{q}(G) = 1 for every prime q ≠ p. If G does not p′-involve Qd(p) when p > 2, then there exists a finite G-CW-complex X ≃ S^{n} with rank one p-group isotropy.

We will obtain Theorem A from a more general technical result, Theorem 6.1, which accepts as input a suitable collection of ℱ_{p}-representations (see Definition 4.1), and produces a finite G-CW complex. Theorem 6.1 is used to construct the action in Example 7.5 for G = A_{7} with rank one p-group isotropy. In principle, it could be used to construct other interesting non-linear examples for finite groups with specified p-group isotropy.

Here is a brief outline of the paper. We denote the orbit category relative to a family ℱ by Γ_{G} = Or_{ℱ} G, and construct projective chain complexes over RΓ_{G} for various p-local coefficient rings R = Z_{(p)}. To prove Theorem 6.1, we first introduce algebraic homotopy representations (see Definition 2.6), as chain complexes over RΓ_{G} satisfying algebraic versions of the conditions found in tom Dieck’s geometric homotopy representations (see [15, II.10.1], [4], [5], and Remark 3.3). In Section 2 we show that the conditions in Definition 2.6 are necessary conditions for a chain complex over RΓ_{G} to be homotopy equivalent to a chain complex of a geometric homotopy representation (see Proposition 2.7). Then in Section 3 we prove conversely that algebraic homotopy representations are realizable by geometric homotopy representations.

In Section 4, we construct p-local chain complexes where the isotropy subgroups are p-groups. In Section 5, we add homology to these local models so that these modified local complexes C^{(p)} all have exactly the same dimension function. Results established in [6] are used to glue these algebraic complexes together over ZΓ_{G}, and then to eliminate a finiteness obstruction. In Section 6 we combine these ingredients to give a complete proof for Theorem 6.1 and Theorem A. We end the paper with a discussion about the necessity of the conditions in Theorem A. This discussion and the examples of nonlinear actions for the groups G = A_{6} and A_{7} can be found in Section 7.

Acknowledgement. The authors would like to thank Alejandro Adem and Assaf Libman for helpful conversations on a number of occasions.

2. Algebraic homotopy representations

Let G be a finite group and ℱ be a family of subgroups of G which is closed under conjugations and taking subgroups. The orbit category Or_{ℱ} G is defined as the category whose objects are orbits of type G/K, with K ∈ ℱ, and where the morphisms from G/K to G/L are given by G-maps:

Mor_{Or_{ℱ} G}(G/K, G/L) = Map_{G}(G/K, G/L).

The category Γ_{G} = Or_{ℱ} G is a small category, and we can consider the module category over Γ_{G}. Let R be a commutative ring with unity. A (right) RΓ_{G}-module M is a contravariant functor from Γ_{G} to the category of R-modules. We denote the Rmodule M(G/K) simply by M(K) and write M(f): M(L) → M(K) for a G-map f: G/K → G/L.

The category of RΓ_{G}-modules is an abelian category, so the usual concepts of homological algebra, such as kernel, direct sum, exactness, projective module, etc., exist for RΓ_{G}-modules. Note that an exact sequence of RΓ_{G}-modules 0 → A → B → C → 0 is exact if and only if

0 → A(K) → B(K) → C(K) → 0

is an exact sequence of R-modules for every K ∈ ℱ. For an RΓ_{G}-module M the Rmodule M(K) can also be considered as an RW_{G}(K)-module in an obvious way where W_{G}(K) = N_{G}(K)/K. We will follow the convention in [11] and consider M(K) as a right RW_{G}(K)-module. In particular, we will consider the sequence above as an exact sequence of right RW_{G}(K)-modules. The further details about the properties of modules over the orbit category, such as the definitions of free and projective modules, can be found in [6] (see also Lück [11, §9,§17] and tom Dieck [15, §10-11]).

In this section we consider chain complexes C of RΓ_{G}-modules. When we say a chain complex we always mean a non-negative complex, i.e., C_{i} = 0 for i < 0. We call a chain complex C projective (resp. free) if for all i ≥ 0, the modules C_{i} are projective (resp. free). We say that a chain complex C is finite if C_{i} = 0 for i > n, and the chain modules C_{i} are all finitely-generated RΓ_{G}-modules.

Given a G-CW-complex X, associated to it, there is a chain complex of RΓ_{G}-modules

C(X^{?}; R) : ⋯ → R[X_{n}^{?} ] ⟶^{∂_{n}} R[X_{n−1}^{?} ] → ⋯ ⟶^{∂_{1}} R[X_{0}^{?} ] → 0

where X_{i} denotes the set of i-dimensional cells in X and R[X_{i}^{?} ] is the RΓ_{G}-module defined by R[X_{i}^{?} ](H) = R[X_{i}^{H}]. We denote the homology of this complex by H_{∗}(X^{?}; R). If the family ℱ includes the isotropy subgroups of X, then the complex C(X^{?}; R) is a chain complex of free RΓ_{G}-modules.

Given a finite dimensional G-CW-complex X, there is a dimension function

Dim X: 𝒮 (G) → Z,

given by (Dim X)(H) = dim X^{H} for all H ∈ 𝒮 (G) where 𝒮 (G) denote the set of all subgroups of G. In a similar way, we define the following.

Definition 2.1. The dimension function of a finite dimensional chain complex C over RΓ_{G} is defined as the function Dim C: 𝒮 (G) → Z given by

(Dim C)(H) = dim C(H)

for all H ∈ ℱ, where the dimension of a chain complex of R-modules is defined as the largest integer d such C_{d} ≠ 0. If C(H) is the zero complex or if H is a subgroup such that H ∉ ℱ, then we define (Dim C)(H) = −1.

Remark 2.2. Recall that a function \underline{n}: 𝒮 (G) → Z is called a super class function if it is constant on conjugacy classes of G. We say that a super class function \underline{n} : 𝒮(G) → Z is defined on ℱ, if \underline{n}(H) = −1 for all subgroups H ∉ ℱ. For such a function, we sometimes use the notation \underline{n} : ℱ → Z instead of \underline{n} : 𝒮 (G) → Z.

In this sense, the dimension function Dim C of a chain complex C over RΓ_{G} is a super class function defined on ℱ. In a similar way, we can define the homological dimension function of a chain complex C of RΓ_{G}-modules as the function HomDim C: ℱ → Z where for each H ∈ ℱ, the integer

(HomDim C)(H) = hdim C(H)

is defined as the homological dimension of the complex C(H).

Let us write (H) ≤ (K) whenever H^{g} ≤ K for some g ∈ G. Here (H) denotes the set of subgroups conjugate to H in G. The notation (H) < (K) means that (H) ≤ (K) but (H) ≠ (K).

Definition 2.3. We call a function \underline{n}: 𝒮 (G) → Z monotone if it satisfies the property that \underline{n}(K) ⩽ \underline{n}(H) whenever (H) ≤ (K). We say that a monotone function \underline{n} is strictly monotone if \underline{n}(K) < \underline{n}(H), whenever (H) < (K). □

We have the following:

Lemma 2.4. The dimension function of a projective chain complex of RΓ_{G}-modules is a monotone function.

Proof. By the decomposition theorem for projective RΓ_{G}-modules [15, Chap. I, Theorem 11.18], every projective RΓ_{G}-module P is of the form P ≅ ⊕_{H}E_{H}P_{H} where P_{H} is a projective N_{G}(H)/H-module. If \underline{n}(K) = n, then C_{n} must have a summand E_{H}P_{H} with (K) ≤ (H). But then we will have C_{n}(L) ≠ 0 for every (L) ≤ (K). Also if H ∉ ℱ, then by the closure of ℱ under taking subgroups, there can not be any K ∈ ℱ with (H) < (K). So we have (Dim C)(K) = −1 for all K ≤ G with (H) ≤ (K). □

We are particularly interested in chain complexes which have the homology of a sphere when evaluated at every K ∈ ℱ. To specify the restriction maps in dimension zero, we will consider chain complexes which are augmented, i.e., chain complexes C together with a map ε: C_{0} → \underline{R} such that ε ∘ ∂_{1} = 0 where \underline{R} denotes the constant functor. We sometimes consider ε as a chain map by considering \underline{R} as a chain complex over RΓ_{G} which is concentrated at zero. By the reduced homology of an augmented complex ε: C → \underline{R}, we always mean the homology of the chain complex

C̃ = \{⋯ → C_{n} ⟶^{∂_{n}} ⋯ → C_{2} ⟶^{∂_{2}} C_{1} ⟶^{∂_{1}} C_{0} ⟶^{ϵ} \underline{R} → 0\}

where \underline{R} is considered to be at dimension −1.

Definition 2.5. Let \underline{n} be a super class function defined on ℱ, and let C be a chain complex over RΓ_{G}.

(i) We say that C an R-homology \underline{n}-sphere if C is an augmented complex with the reduced homology of C(K) is the same as the reduced homology of an \underline{n}(K)- sphere (with coefficients in R) for all K ∈ ℱ.

(ii) We say that C is oriented if the W_{G}(K)-action on the homology of C(K) is trivial for all K ∈ ℱ.

In transformation group theory, a G-CW-complex X is called a homotopy representation if it has the property that X^{H} is homotopy equivalent to the sphere S^{n(H)} where n(H) = dim X^{H} for every H ≤ G (see tom Dieck [15, Section II.10]). Note that we do not assume that the dimension function is strictly monotone as in Definition II.10.1 in [15].

In [15, II.10], there is a list of properties that are satisfied by homotopy representations. We will use algebraic versions of these properties to define an analogous notion for chain complexes.

Definition 2.6. Let C be a finite projective chain complex over RΓ_{G}, which is an Rhomology \underline{n}-sphere. We say C is an algebraic homotopy representation (over R) if

(i) The function \underline{n} is a monotone function.

(ii) If H, K ∈ ℱ are such that n = \underline{n}(K) = \underline{n}(H), then for every G-map f: G/H → G/K the induced map C(f): C(K) → C(H) is an R-homology isomorphism.

(iii) Suppose H, K, L ∈ ℱ are such that H ≤ K, L and let M = ⟨K, L⟩ be the subgroup of G generated by K and L. If n = \underline{n}(H) = \underline{n}(K) = \underline{n}(L) > −1, then M ∈ ℱ and n = \underline{n}(M).

In the remainder of this section we will assume that R is a principal ideal domain. The main examples for us are R = Z_{(p)} or R = Z.

Proposition 2.7. Let C be a finite projective chain complex over RΓ_{G}, which is an Rhomology \underline{n}-sphere. If the equality \underline{n} = Dim C holds, then C is an algebraic homotopy representation.

Under condition (iii) of Definition 2.6, the isotropy family ℱ has an important maximality property.

Corollary 2.8. Let C be a projective chain complex of RΓ_{G}-modules, If condition (iii) holds, then the set of subgroups ℱ_{H} = \{K ∈ ℱ | (H) ≤ (K), \underline{n}(K) = \underline{n}(H) > −1\} has a unique maximal element, up to conjugation.

Before we prove Proposition 2.7, we make some observations and give some definitions for projective chain complexes.

Lemma 2.9. Let C be a projective chain complex over RΓ_{G}. Then, for every G-map f: G/H → G/K, the induced map C(f): C(K) → C(H) is an injective map with an R-torsion free cokernel.

Proof. It is enough to show that if P a projective RΓ_{G}-module, then for every G-map f: G/H → G/K, the induced map P (f): P (K) → P (H) is an injective map with a torsion free cokernel. Since every projective module is a direct summand of a free module, it is enough to prove this for a free module P = R[X^{?}]. Let f: G/H → G/K be the Gmap defined by f(H) = gK. Then the induced map P (f): R[X^{K}] → R[X^{H}] is the linearization of the map X^{K} → X^{H} given by x ↦ gx. Since this map is one-to-one, we can conclude that P (f) is injective with torsion free cokernel. □

When H ≤ K and f: G/H → G/K is the G-map defined by f(H) = K, then we denote the induced map C(f): C(K) → C(H) by r_{H}^{K} and call it the restriction map. When H and K are conjugate, i.e., K = H^{g} for some g ∈ G, then the map C(f): C(K) → C(H) induced by the G-map f: G/H → G/K defined by f(H) = gK is called the conjugation map and usually denoted by c_{K}^{g}. Note that every G-map can be written as a composition of two G-maps of the above two types, so every induced map C(f): C(K) → C(H) can be written as a composition of restriction and conjugation maps.

Since conjugation maps have inverses, they are always isomorphisms. So, the condition (ii) of Definition 2.6 is actually a statement only about restriction maps. To study the restriction maps more closely, we consider the image of r_{H}^{K}: C(K) → C(H) for a pair H ≤ K and denote it by C_{H}^{K}. Note that C_{H}^{K} is a subcomplex of C(H) as a chain complex of R-modules. Also note that if C is a projective chain complex, then C_{H}^{K} is isomorphic to C(K), as a chain complex of R-modules, by Lemma 2.9.

Lemma 2.10. Let C be a projective chain complex over RΓ_{G}. Suppose that K, L ∈ ℱ such that H ≤ K and H ≤ L, and let M = ⟨K, L⟩ be the subgroup generated by K and L. If C_{H}^{K} ∩ C_{H}^{L} ≠ 0 then M ∈ ℱ and we have C_{H}^{K} ∩ C_{H}^{L} = C_{H}^{M}.

Proof. As before it is enough to prove this for a free RΓ_{G}-module P = R[X^{?}] where X is a G-set whose isotropy subgroups lie in ℱ. Note that the restriction maps r_{H}^{K} and r_{H}^{L} are linearizations of the maps X^{K} → X^{H} and X^{L} → X^{H}, respectively, which are defined by inclusion of subsets. Then it is clear that the intersection of images of r_{H}^{K} and r_{H}^{L} would be R[X^{K} ∩ X^{L}] considered as an R-submodule of R[X^{H}]. There is a well known equality X^{K} ∩ X^{L} = X^{M} for fixed point sets. Therefore, if C_{H}^{K} ∩ C_{H}^{L} ≠ 0, then we must have X^{M} ≠ ∅. This implies that M ∈ ℱ and that im r_{H}^{K} ∩ im r_{H}^{L} = im r_{H}^{M}. □

Now, we are ready to prove Proposition 2.7.

Proof of Proposition 2.7. The first condition in Definition 2.6 follows from Lemma 2.4. For (ii) and (iii), we use the arguments similar to the arguments given in II.10.12 and II.10.13 in [15].

To prove (ii), let f: G/H → G/K be a G-map. By Lemma 2.9, the induced map C(f) : C(K) → C(H) is injective with torsion free cokernel. Let D denote the cokernel of C(f). Then we have a short exact sequence of R-modules

0 → C(K) → C(H) → D → 0

where both C(K) and C(H) have dimension n. Now consider the long exact reduced homology sequence (with coefficients in R) associated to this short exact sequence:

⋯ → 0 → H_{n+1}(D) → H_{n}(C(K)) ⟶^{f^{∗}} H_{n}(C(H)) → H_{n}(D) → ⋯

Note that D has dimension less than or equal to n, so H_{n+1}(D) = 0 and H_{n}(D) is torsion free. Since H_{n}(C(K)) = H_{n}(C(H)) = R, we obtain that f^{∗} is an isomorphism. Since both C(K) and C(H) have no other reduced homology, we conclude that C(f) induces an R-homology isomorphism between associated augmented complexes. Since the induced map \underline{R}(f) : \underline{R}(K) → \underline{R}(H) is the identity map id : R → R, the chain map C(f): C(L) → C(K) is an R-homology isomorphism.

To prove (iii), observe that there is a Mayer-Vietoris type exact sequence associated to the pair of complexes C_{H}^{K} and C_{H}^{L} which gives an exact sequence of the form

0 → H_{n}(C_{H}^{K} ∩ C_{H}^{L}) → H_{n}(C_{H}^{K}) ⊕ H_{n}(C_{H}^{L}) → H_{n}(C_{H}^{K} + C_{H}^{L}) → H_{n−1}(C_{H}^{K} ∩ C_{H}^{L}) → 0.

Here we again take the homology sequence as the reduced homology sequence.

Let i^{K}: C_{H}^{K} → C(H), i_{H}^{L}: C_{H}^{L} → C(H), and j: C_{H}^{K} +C_{H}^{L} → C(H) denote the inclusion maps. We have zero on the left-most term since C_{H}^{K} + C_{H}^{L} is an n-dimensional complex. To see the zero on the right-most term, note that by Lemma 2.9, C_{H}^{K} ≅ C(K) and C_{H}^{L} ≅ C(L) as chain complexes of R-modules, so they have the same homology. This gives that H_{i}(C_{H}^{K}) = H_{i}(C_{H}^{L}) = 0 for i ≤ n − 1.

Also note that by part (ii), the composition

H_{n}(C(K)) ≅ H_{n}(C_{H}^{K}) ⟶^{i_{∗}^{K}} H_{n}(C_{H}^{K} + C_{H}^{L}) ⟶^{j_{∗}} H_{n}(C(H))

is an isomorphism. So, j_{∗} is surjective. Since H_{n+1}(C(H)/(C_{H}^{K} + C_{H}^{L})) = 0, we see that j_{∗} is also injective. Therefore, j_{∗} is an isomorphism. This implies that i_{∗}^{K} is an isomorphism. Similarly one can show that i_{∗}^{L}: H_{n}(C_{H}^{L}) → H_{n}(C_{H}^{K} + C_{H}^{L}) is also an isomorphism. Using these isomorphisms and looking at the exact sequence above, we conclude that H_{n}(C_{H}^{K} ∩ C_{H}^{L}) ≅ R and H_{i}(C_{H}^{K} ∩ C_{H}^{L}) = 0 for i ≤ n − 1. So, C_{H}^{K} ∩ C_{H}^{L} is an R-homology n-sphere.

Since n > −1, this implies that C_{H}^{K} ∩ C_{H}^{L} ≠ 0, and hence M = ⟨K, L⟩ ∈ ℱ by Lemma 2.10. Moreover, C_{H}^{K} ∩ C_{H}^{L} = C_{H}^{M}. This proves that \underline{n}(M) = n as desired. □

3. A Realization Theorem for Chain Complexes over ZΓ_{G}

In [6], we proved the following realization theorem for free ZΓ_{G}-module chain complexes, with respect to any family ℱ, which are Z-homology \underline{n}-spheres satisfying certain extra conditions.

Theorem 3.1 ([6, Theorem 8.10], [12]). Let C be a finite free chain complex of ZΓ_{G}- modules which is a homology \underline{n}-sphere. Suppose that \underline{n}(K) ≥ 3 for all K ∈ ℱ. If C_{i}(H) = 0 for all i > \underline{n}(H) + 1, and all H ∈ ℱ, then there is a finite G-CW-complex X such that C(X^{?}; Z) is chain homotopy equivalent to C as chain complexes of ZΓ_{G}-modules.

Note that a homology \underline{n}-sphere C with Dim C = \underline{n}, and \underline{n}(K) ≥ 3 for all K ∈ ℱ, will automatically satisfy these conditions. So, it is interesting to ask under what conditions a projective chain complex C which is an Z-homology \underline{n}-sphere is chain homotopy equivalent to one where the equality Dim C = \underline{n} holds. It turns out that the conditions (i), (ii), and (iii) of Definition 2.6 are exactly what is needed. In other words, the additional input needed is that C should be an algebraic homotopy representation.

The main purpose of this section is prove the following theorem.

Theorem 3.2. Let C be a finite free chain complex of ZΓ_{G}-modules which is a homology \underline{n}-sphere. If C is an algebraic homotopy representation, then C is chain homotopy equivalent to a finite free chain complex D for which Dim D = \underline{n}. If, in addition, \underline{n}(K) ≥ 3 for all K ∈ ℱ, then there is a finite G-CW-complex X such that C(X^{?}; Z) is chain homotopy equivalent to C as chain complexes of ZΓ_{G}-modules.

Remark 3.3. The construction actually produces a finite G-CW-complex X with the additional property that all the non-empty fixed sets X^{H} are simply-connected. Moreover, by construction, W_{G}(H) = N_{G}(H)/H will act trivially on the homology of X^{H}. Therefore X will be an oriented geometric homotopy representation (in the sense of tom Dieck). From the perspective of Theorem A, since we don’t specify any dimension function, a G-CW-complex X with all fixed sets X^{H} integral homology spheres will lead (by threefold join) to a homotopy representation. The same necessary and sufficient conditions for existence apply.

The last sentence of Theorem 3.2 follows from Theorem 3.1. The first part says that under the conditions given in the theorem, the complex C is homotopy equivalent to a finite free chain complex D with Dim D(H) = HomDim D(H) = \underline{n}(H) for every H ∈ ℱ.

In the remainder of this section we will again assume that R is a principal ideal domain. The main examples for us are R = Z_{(p)} or R = Z, as before.

Definition 3.4. We say a chain complex C of RΓ_{G}-modules is tight at H ∈ ℱ if

Dim C(H) = HomDim C(H).

We call a chain complex of RΓ_{G}-modules tight if it is tight at every H ∈ ℱ.

For possible future applications we will deduce Theorem 3.2 from the following algebraic result, together with Theorem 3.1.

Theorem 3.5. Let C be a finite free chain complex of RΓ_{G}-modules which is a homology \underline{n}-sphere. If C is an algebraic homotopy representation over R, then C is chain homotopy equivalent to a finite free chain complex D which is tight.

So to prove Theorem 3.5, we need to show that the complex C can be made tight at each H ∈ ℱ by replacing it with a chain complex homotopic to C.

Let H be a maximal element in ℱ. Consider the subcomplex C^{(H)} of C formed by free summands of C isomorphic to R[G/H^{?} ]. The complex C^{(H)} is a complex of isotypic modules of type R[G/H^{?} ]. Recall that free RΓ_{G}-module F is called isotypic of type G/H if it is isomorphic to a direct sum of copies of a free module R[G/H^{?} ], for some H ∈ ℱ. For extensions involving isotypic modules we have the following:

Lemma 3.6. Let

ℰ: 0 → F → F′ → M → 0

be a short exact sequence of RΓ_{G}-modules such that both F and F′ are isotypic free modules of the same type G/H. If M(H) is R-torsion free, then ℰ splits and M is stably free.

Proof. This is Lemma 8.6 of [6]. The assumption that R is a principal ideal domain ensures that finitely-generated R-torsion free modules are free. □

Note that C^{(H)}(H) = C(H), since H is maximal in ℱ. This means that C^{(H)} is a finite free chain complex over RΓ_{G} of the form

C^{(H)} : 0 → F_{d} → F_{d−1} → ⋯ → F_{1} → F_{0} → 0

which is a R-homology \underline{n}(H)-sphere, with \underline{n}(H) ≤ d.

Lemma 3.7. Let C be a finite free chain complex of RΓ_{G}-modules. Then C is chain homotopy equivalent to a finite free chain complex D which is tight at every maximal element H ∈ ℱ.

Proof. We apply [6, Proposition 8.7] to the subcomplex C^{(H)}, for each maximal element H ∈ ℱ. The key step is provided by Lemma 3.6. □

To make the complex C tight at every H ∈ ℱ we use a downward induction, but the situation at an intermediate step is more complicated than the first step considered above.

Suppose that H ∈ ℱ is such that C tight at every K ∈ ℱ such that (K) > (H). Let C^{H} denote the subcomplex of C with free summands of type R[G/K^{?} ] satisfying (H) ≤ (K). In a similar way, we can define the subcomplex C^{>H} of C whose free summands are of type R[G/K^{?} ] with (H) < (K). The complex C^{>H} is a subcomplex of C^{H}. Let us denote the quotient complex C^{H}/C^{>H} by C^{(H)}. As before the complex C^{(H)} is isotypic with isotropy type R[G/H^{?} ]. We have a short exact sequence of chain complexes of free RΓ_{G}-modules

0 → C^{>H} → C^{H} → C^{(H)} → 0.

By evaluating at H, we obtain an exact sequence of chain complexes

0 → C^{>H}(H) → C^{H}(H) → C^{(H)}(H) → 0

which is just the sequence

0 → C^{>H}(H) → C(H) → S_{H}C → 0

defining the splitting functor S_{H} (see [11, Lemma 9.26]) Note that we also have a sequence

0 → C^{H} → C → C/C^{H} → 0.

If we can show that C^{H} is homotopy equivalent to a complex D′ which is tight at H, then by push-out of D′ along the injective map C^{H} → C, we can find a complex D homotopy equivalent to C which is tight at every K ∈ ℱ with (H) ≤ (K). So it is enough to show that C^{H} is homotopy equivalent to a complex D′ which is tight at H.

Lemma 3.8. Let C be a finite free chain complex of RΓ_{G}-modules which is tight at every K ∈ ℱ with (H) < (K). Suppose that n = hdim C(H) ≥ dim C(K) for all (H) < (K) and that H_{n+1}(S_{H}C) = 0. Then C^{H} is homotopy equivalent to a finite free chain complex D′ which is tight at every K ∈ ℱ with (H) ≤ (K).

Proof. We first observe that C^{>H} has dimension ≤ n, since C^{>H}(K) = C(K) for (H) < (K), and dim C(K) ≤ n. Let d = dim C(H). If d = n, then we are done, so assume that d > n. Then dim C^{(H)} = d, and C^{(H)} is a complex of the form

C^{(H)} : 0 → F_{d} → F_{d−1} → ⋯ → F_{1} → F_{0} → 0.

We claim that the first map in the above chain complex is injective. Note that since C^{(H)} is isotypic of type (H), it is enough to show that this map is injective when it is calculated at H. In other words we claim that H_{d}(C^{(H)}(H)) = H_{d}(S_{H}C) = 0 when d > n. To show this consider the short exact sequence 0 → C^{>H}(H) → C(H) → S_{H}C → 0. Since the complex C^{>H} has dimension ≤ n, the corresponding long exact sequence gives that H_{d}(S_{H}C) ≅ H_{d}(C(H)) = 0 when d > n + 1. If d = n + 1, then this is true by the assumption in the lemma. Now we apply [6, Proposition 8.7] to C^{(H)} to obtain a tight complex D′′ ≃ C^{(H)}, and then let D′ ≃ C^{H} denote the pullback of D′′ along the surjection C^{H} → C^{(H)}. □

To complete the proof of Theorem 3.5, we need to show that if C is a finite free chain complex of RΓ_{G}-modules which is an R-homology \underline{n}-sphere, satisfying the conditions (i), (ii), and (iii) in Definition 2.6, then the conditions in Lemma 3.8 hold at an intermediate step of the downward induction.

Note that the condition about the dimensions follows from the condition (i) since when \underline{n} is monotone, we have

HomDim C(H) = \underline{n}(H) ≥ \underline{n}(K) = HomDim C(K) = Dim(K)

for all K ∈ ℱ with (H) < (K). The other condition in the lemma, that H_{n+1}(S_{H}C) = 0, follows from the conditions (ii) and (iii) but it is more involved to show this (see Corollary 3.11 below).

In the rest of the section, we assume that C is a finite projective chain complex of RΓ_{G}-modules, which is an R-homology \underline{n}-sphere, and satisfies the conditions (i), (ii), and (iii) in Definition 2.6. Assume also that C is tight for all K ∈ ℱ with (H) < (K) for some fixed subgroup H ∈ ℱ. Let 𝒦_{H} denote the set of all subgroups K ∈ ℱ such that H < K and \underline{n}(H) = \underline{n}(K). By condition (iii) of Definition 2.6, this collection has a unique maximal element M. Let C_{H}^{K} denote the image of the restriction map

r_{H}^{K}: C(K) → C(H),

for every K ∈ ℱ with H ≤ K. Note that C_{H}^{K} is a subcomplex of C(H) and by Lemma 2.9, it is isomorphic to C(K). Moreover, if K ∈ 𝒦_{H}, then by condition (ii), the subcomplex C_{H}^{K} is an R-homology n-sphere and the map

H_{n}(C_{H}^{M}) → H_{n}(C_{H}^{K})

induced by the inclusion map C_{H}^{M} ↪ C_{H}^{K} is an isomorphism. More generally, the following also holds.

Lemma 3.9. Let C and H ∈ ℱ be as above, and let K_{1}, …, K_{m} be a set of subgroups in 𝒦_{H}. Then the subcomplex ∑_{i=1}^{m} C_{H}^{K_{i}} is an R-homology n-sphere and the map

(3.10)  H_{n}(C_{H}^{M}) → H_{n}(∑_{i=1}^{m} C_{H}^{K_{i}})

induced by the inclusion maps is an isomorphism.

Proof. The case m = 1 follows from the remarks above. For m > 1, we have the following Mayer-Vietoris type long exact sequence

0 → H_{n}(D_{m−1} ∩ C_{H}^{K_{m}}) → H_{n}(D_{m−1}) ⊕ H_{n}(C_{H}^{K_{m}}) → H_{n}(D_{m}) → H_{n−1}(D_{m−1} ∩ C_{H}^{K_{m}}) →

where D_{j} = ∑_{i=1}^{j} C_{H}^{K_{i}} for j = m − 1, m. By the inductive assumption, we know that D_{m−1} is an R-homology n-sphere and the map H_{n}(C_{H}^{M}) → H_{n}(D_{m−1}) is an isomorphism.

Note that

D_{m−1} ∩ C_{H}^{K_{m}} = (∑_{i=1}^{m−1} C_{H}^{K_{i}}) ∩ C_{H}^{K_{m}} = ∑_{i=1}^{m−1}(C_{H}^{K_{i}} ∩ C_{H}^{K_{m}}) = ∑_{i=1}^{m−1} C_{H}^{⟨K_{i},K_{m}⟩}

where the last equality follows from Lemma 2.10. We can apply Lemma 2.10 here because C_{H}^{M} ⊆ C_{H}^{K} for all K ∈ 𝒦_{H} gives that C_{H}^{K_{i}} ∩ C_{H}^{K_{m}} ≠ 0 for every i = 1, …, m − 1. Note that we also obtain ⟨K_{i}, K_{m}⟩ ∈ 𝒦_{H} for all i. Applying our inductive assumption again to these subgroups, we obtain that D_{m−1} ∩ C_{H}^{K_{m}} is an R-homology n-sphere and that the induced map

H_{n}(C_{H}^{M}) → H_{n}(D_{m−1} ∩ C_{H}^{K_{m}})

is an isomorphism. This gives that H_{i}(D_{m}) = 0 for i ≤ n−1. We also obtain a commuting diagram

\xymatrix{0 \ar[r] & H_{n}(C_{H}^{M}) \ar[r] \ar[d] & H_{n}(C_{H}^{M}) ⊕ H_{n}(C_{H}^{M}) \ar[r] \ar[d] & H_{n}(C_{H}^{M}) \ar[r] \ar[d]^{φ} & 0 \\ 0 \ar[r] & H_{n}(D_{m−1} ∩ C_{H}^{K_{m}}) \ar[r] & H_{n}(D_{m−1}) ⊕ H_{n}(C_{H}^{K_{m}}) \ar[r] & H_{n}(D_{m}) \ar[r] & 0}

Since all the vertical maps except the map φ are known to be isomorphisms, we obtain that φ is also an isomorphism by the five lemma. This completes the proof. □

Corollary 3.11. Let C and H ∈ ℱ are as above. Then H_{n+1}(S_{H}C) = 0.

Proof. Let 𝒦_{H} = {K_{1}, …, K_{m}}. By condition (ii), we know that the composition

H_{n}(C(M)) ⟶^{≅} H_{n}(C_{H}^{M}) → H_{n}(∑_{i=1}^{m} C_{H}^{K_{i}}) → H_{n}(C(H))

is an isomorphism. However, we have just proved that the middle map is an isomorphism, and that all the modules involved in the composition are isomorphic to R. Therefore, the map induced by inclusion

H_{n}(∑_{i=1}^{m} C_{H}^{K_{i}}) → H_{n}(C(H))

is an isomorphism. Note that if (H) ≤ (K) and \underline{n}(K) < n, for some K ∈ ℱ, then dim C(K) < n. This means that

H_{n}(C^{>H}(H)) ≅ H_{n}(∑_{i=1}^{m} C_{H}^{K_{i}}) ≅ H_{n}(C(H)).

From the exact sequence 0 → C^{>H}(H) → C(H) → S_{H}C → 0, and the fact that HomDim C(H) = n, we conclude that H_{n+1}(S_{H}C) = 0, as required. □

4. Construction of the preliminary local models

Our main technical tool is provided by Theorem 6.1, which gives a method for constructing finite G-CW-complexes, with isotropy in a given family. This theorem will be proved by applying the realization statement of Theorem 3.2. To construct a suitable finite free chain complex C over ZΓ_{G}, we work one prime at a time to construct local models C^{(p)}, and then apply the glueing method for chain complexes developed in [6, Theorem 6.7].

The main input of Theorem 6.1 is a compatible collection of unitary representations for the p-subgroups of G. We give the precise definition in a more general setting.

Definition 4.1. Let ℱ be a family of subgroups of G. We say that V(ℱ) is an ℱrepresentation for G, if V(ℱ) = {V_{H} ∈ Rep(H, U(n)) | H ∈ ℱ} is a compatible collection of (non-zero) unitary H-representations. The collection is compatible if f^{∗}(V_{K}) ≅ V_{H} for every G-map f: G/H → G/K.

For any finite G-CW-complex X, we let Iso(X) = {K ≤ G | X^{K} ≠ ∅} denote the isotropy family of the G-action on X. This suggests the following notation:

Definition 4.2. Let V(ℱ) be an ℱ-representation for G. We let

Iso(V(ℱ)) = {L ≤ H | S(V_{H})^{L} ≠ ∅, for some V_{H} ∈ V(ℱ)}

denote the isotropy family of V(ℱ). We note that Iso(V(ℱ)) is a sub-family of ℱ.

Example 4.3. Our first example for these definitions will be a compatible collection of representations for the family ℱ_{p} of all p-subgroups, with p a fixed prime dividing the order of G. In this case, an ℱ_{p}-representation V(ℱ_{p}) can be constructed from a suitable representation V_{p} ∈ Rep(P, U(n)), where P denotes a p-Sylow subgroup of G. The representations V_{H} can be constructed for all H ∈ ℱ_{p}, by extending V_{p} to conjugate p-Sylow subgroups and by restriction to subgroups. To ensure a compatible collection {V_{H}}, we assume that V_{p} respects fusion in G, meaning that χ_{p}(gxg^{−1}) = χ_{p}(x) for the corresponding character χ_{p}, whenever gxg^{−1} ∈ P for some g ∈ G and x ∈ P.

We will now specify an isotropy family 𝒥 that will be used in the rest of the paper.

Definition 4.4. Let {V(ℱ_{p}) | p ∈ 𝒮_{G}} be a collection of ℱ_{p}-representations, for a set 𝒮_{G} of primes dividing the order of G. Let 𝒥_{p} = Iso(V(ℱ_{p})) and 𝒥 = ⋃{𝒥_{p} | p ∈ 𝒮_{G}} denote the isotropy families.

We note that 𝒥 contains no isotropy subgroups of composite order, since each 𝒥_{p} is a family of p-subgroups. Let Γ_{G} = Or_{𝒥} G and Γ_{G}(p) denote the orbit category Or_{𝒥_{p}} G over the family 𝒥_{p}. A chain complex C over RΓ_{G}(p) can always be considered as a complex of RΓ_{G}-modules, by taking the values C(H) at subgroups H ∉ 𝒥_{p} as zero complexes.

In this section we construct a p-local chain complex C^{(p)}(0) over RΓ_{G}(p), for R = Z_{(p)}, which we call a preliminary local model (see Definition 4.10). From this construction we will obtain a dimension function \underline{n}^{(p)} : 𝒥_{p} → Z. By taking joins we can assume that these dimension functions have common value at H = 1. In the next section, these preliminary local models will be “improved” at each prime p by adding homology as specified by the dimension functions \underline{n}^{(q)} : 𝒥_{q} → Z, for all q ∈ 𝒮_{G} with q ≠ p. The resulting complexes C^{(p)} over the orbit category RΓ_{G} will all have the same dimension function

\underline{n} = ⋃ \{\underline{n}^{(p)} | p ∈ 𝒮_{G}\}: 𝒥 → Z,

and satisfy conditions needed for the glueing method.

The first step is based on a construction of Connolly and Prassidis [2, §2].

Theorem 4.5. Let G be a finite group. Suppose that:

(i) B is a finite-dimensional G-CW-complex, with fixed sets B^{H} simply-connected for all H ≤ G;

(ii) V(ℱ) is an ℱ-representation for G, with ℱ = Iso(B). Then there exists a finite-dimensional G-CW-complex E, with isotropy Iso(V(ℱ)), such that all fixed sets E^{H} are simply-connected.

Proof. The given G-CW-complex B and the given ℱ-representation V(ℱ) can be combined with a construction of Connolly and Prassidis [2, §2] to produce an integer k ≥ 1 and a G-fibration π: E → B, such that the fibre

π^{−1}(σ) ≃_{H} S(V_{H}^{⊕k}),

for all cells σ ⊂ B with stabilizer G_{σ} = H. The construction given in [2, §2] only works when all the representations V_{H} are free H-representations, but this construction later generalized by Klaus [10, Proposition 2.7] to an arbitrary family of unitary representations. A more detailed version of this construction can also be found in [17, Proposition 4.3] (see also Ilhan [7]). Here V_{H}^{⊕k} denotes the direct sum V_{H} ⊕ ⋯ ⊕ V_{H} of k copies of the unitary representation V_{H}, and ≃_{H} denotes “H-equivariant homotopy equivalence”. The G-CW-complex E has the required properties. □

The basic input for our local models is the construction of finite-dimensional G-CW-complexes with good p-local properties.

Corollary 4.6. Let G be a finite group, and let V(ℱ_{p}) be an ℱ_{p}-representation for G for some p ∈ 𝒮_{G}. Then there exists a finite-dimensional G-CW-complex E, which is p-locally homotopy equivalent to a sphere, with isotropy 𝒥_{p} = Iso(V(ℱ_{p})). Moreover, all the fixed sets E^{H} are simply-connected.

Proof. We recall a result of Jackowski, McClure and Oliver [8, Proposition 2.2]: there exists a simply-connected, finite dimensional G-CW-complex B which is F_{p}-acyclic and has finitely many orbit types with isotropy in the family of p-subgroups ℱ_{p} in G. The quoted result applies more generally to all compact Lie groups and produces a complex with p-toral isotropy (meaning a compact Lie group P whose identity component P_{0} is a torus, and P/P_{0} is a finite p-group). For G finite, the p-toral subgroups are just the p-subgroups. The property that all fixed sets B^{H} are simply-connected is established in the proof.

We now apply Theorem 4.5 to this G-CW-complex B and the given ℱ_{p}-representation V(ℱ_{p}). The resulting G-CW-complex E has the required properties. In particular, since B is F_{p}-acyclic then for each p-subgroup H, the fixed point set B^{H} will be also F_{p}-acyclic (and B^{H} ≠ ∅). Hence E is a p-local sphere. □

We now let R = Z_{(p)}, and consider the finite dimensional chain complex C(E^{?}; R) of free RΓ_{G}(p)-modules. By taking joins, we may assume that this complex has “homology gaps” of length > l(Γ_{G}) as required for [6, Theorem 6.7], and that all the non-empty fixed sets E^{H} have \underline{n}(H) ≥ 3 and trivial action of W_{G}(H) on homology. Let \underline{n}^{(p)} : 𝒥_{p} → Z denote the dimension function HomDim C(E^{?}; R).

The following result applies to chain complexes over RΓ_{G} with respect to any family ℱ of subgroups.

Lemma 4.7. Let R be a noetherian ring and G be a finite group. Suppose that C is an n-dimensional chain complex of projective RΓ_{G}-modules with finitely generated homology groups. Then C is chain homotopy equivalent to a finitely-generated projective n-dimensional chain complex over RΓ_{G}.

Proof. Note that the chain modules of C are not assumed to be finitely-generated, but H_{i}(C) = 0 for i > n. We first apply Dold’s “algebraic Postnikov system” technique [3, §7], to chain complexes over the orbit category (see [6, §6]).

According to this theory, given a positive projective chain complex C, there is a sequence of positive projective chain complexes C(i) indexed by positive integers such that f: C → C(i) induces a homology isomorphism for dimensions ≤ i. Moreover, there is a tower of maps

\xymatrix{ & C(i) \ar[d] \\  & C(i − 1) \ar@{-->}[d] \ar[r]^{α_{i}} & Σ^{i+1}P(H_{i}) \\ C \ar[dr] \ar[ur] \ar[uur] \ar[r] & C(1) \ar[d] \ar[r]^{α_{2}} & Σ^{3}P(H_{2}) \\  & C(0) \ar[r]^{α_{1}} & Σ^{2}P(H_{1})}

such that C(i) = Σ^{−1}C(α_{i}), where C(α_{i}) denotes the algebraic mapping cone of α_{i}, and P(H_{i}) denotes a projective resolution of the homology module H_{i} = H_{i}(C).

By assumption, since the homology modules H_{i} are finitely-generated and R is noetherian, we can choose the projective resolutions P(H_{i}) to be finitely-generated in each degree. Therefore, at each step the chain complex C(i) consists of finitely-generated projective RΓ_{G}-modules, and C(n) ≃ C has homological dimension ≤ n. Now, since H^{n+1}(C(n); M) = H^{n+1}(C; M) = 0, for any RΓ_{G}-module M, we conclude that C(n) is chain homotopy equivalent to an n-dimensional finitely-generated projective chain complex by [11, Prop. 11.10]. □

Remark 4.8. This result generalizes [11, 11.31:ex. 2] or [14, Satz 9].

Lemma 4.9. The chain complex C(E^{?}; R) is chain homotopy equivalent to an oriented R-homology \underline{n}^{(p)}-sphere C^{(p)}(0), which is an algebraic homotopy representation.

Proof. The chain complex C(E^{?}; R) is finite dimensional and free over RΓ_{G}, but may not be finitely-generated. However, by the conclusion of Corollary 4.6, the homology groups H_{∗}(C(E^{?}; R)) are finitely generated since C(E^{?}; R) is an R-homology \underline{n}-sphere. The result now follows from Lemma 4.7, which produces a finite length projective chain complex C^{(p)}(0) of finitely-generated RΓ_{G}(p)-modules. Note that C(E^{?}; R) satisfies the conditions (i)-(iii) in Definition 2.6, so C^{(p)}(0) also satisfies these conditions (which are chain-homotopy invariant), hence C^{(p)}(0) is an algebraic homotopy representation. □

Note that C^{(p)}(0) is an algebraic homotopy representation, meaning that it satisfies the condition (i), (ii), and (iii) in Definition 2.6, even though Dim C^{(p)} may not be equal to \underline{n}^{(p)} = HomDim C^{(p)}(0).

By taking joins, we may assume that there exists a common dimension N = \underline{n}^{(p)}(1), at H = 1, for all all p ∈ 𝒮_{G}. Moreover, we may assume that N + 1 is a multiple of any given integer m_{G} (to be chosen below). We now obtain the “global” dimension function

\underline{n} = ⋃ \{\underline{n}^{(p)} | p ∈ 𝒮_{G}\}: 𝒥 → Z,

where \underline{n}^{(p)} = HomDim C^{(p)}(0), for all p ∈ 𝒮_{G}, and \underline{n}(1) = N.

Definition 4.10 (Preliminary local models). Let 𝒮_{G} = {p | rank_{p} G ≥ 2}, and let m_{G} denote the least common multiple of the q-periods for G (as defined in [13, p. 267]), over all primes q for which rank_{q} G = 1. We assume that \underline{n}(1) + 1 is a multiple of m_{G}.

(i) We will take the chain complex C^{(p)}(0) constructed in Lemma 4.9 for our preliminary model at each prime p ∈ 𝒮_{G}.

(ii) If rank_{q} G = 1, we take 𝒥_{q} = {1} and C^{(q)}(0) as the RΓ_{G}-chain complex E_{1}P where P is a periodic resolution of R as a RG-module with period \underline{n}(1) + 1 (for more details, see the proof of Theorem 5.2 below, or [6, Section 9B]).

This completes the construction of the preliminary local models at each prime dividing the order of G, for a given family of ℱ_{p}-representations. In the next section we will modify these preliminary models to get p-local chain complexes C^{(p)} over RΓ_{G} which are R-homology \underline{n}-spheres for the dimension function \underline{n} described above.

Example 4.11. In the proof of Theorem A we will be using the setting of Example 4.3. Suppose that G is a rank two finite group which does not p′-involve Qd(p), for any odd prime p. We let 𝒮_{G} be the set of primes p where rank_{p} G = 2. Under this condition, a result of Jackson [9, Theorem 47] asserts that G admits a p-effective p-local character V_{p}. Here p-effective means that when V_{p} is restricted to an elementary abelian subgroup E of rank 2 then it has no trivial summand. This guarantees that the set of isotropy subgroups 𝒥_{p} = Iso(S(V_{p})) consists of the rank one p-subgroups. In this setting, our preliminary local models arise from the following special case:

Corollary 4.12. Let G be a finite rank two group with rank_{p} G = 2. If G does not p′-involve Qd(p) when p > 2, then there exists a simply-connected, finite-dimensional G-CW-complex E with rank one p-group isotropy, which is p-locally homotopy equivalent to a sphere.

5. Construction of the local models: adding homology

Let G be a finite group and let 𝒮_{G} = {p | rank_{p} G ≥ 2}. We will use the notation 𝒥_{p} = Iso(V(ℱ_{p})), for p ∈ 𝒮_{G}, as given in Definition 4.4. For p ∉ 𝒮_{G} we have 𝒥_{p} = {1}. We will continue to work over the orbit category Γ_{G} = Or_{𝒥} G where 𝒥 = ⋃{𝒥_{p} | p ∈ 𝒮_{G}}. For each prime p dividing the order of G, let C^{(p)}(0) denote the preliminary p-local model given in Definition 4.10, and denote the homological dimension function of C^{(p)}(0) by \underline{n}^{(p)} : 𝒥_{p} → Z for all primes dividing the order of G.

We now fix a prime q dividing the order of G, and let R = Z_{(q)}. In Theorem 5.2, we will show how to add homology to the preliminary local model C^{(q)}(0), to obtain an algebraic homotopy representation with dimension function \underline{n}^{(p)} ∪ \underline{n}^{(q)} for any prime p ∈ 𝒮_{G} such that p ≠ q. After finitely many such steps, we will obtain our local model C^{(q)} over RΓ_{G} with dimension function HomDim C^{(q)} = \underline{n}.

Remark 5.1. In order to carry out the above construction, certain conditions on the dimension function must hold. In particular, if there exists a q-local model C^{(q)} with isotropy in 𝒥_{p} ∪ 𝒥_{q}, where p ∈ 𝒮_{G}, then for every p-subgroup 1 ≠ H ∈ 𝒥_{p}, we can consider the RN_{G}(H)/H complex C^{(q)}(H). This is a finite length chain complex of finitely generated modules which has the R-homology of an \underline{n}(H)-sphere.

Since R = Z_{(q)}, if we take a q-subgroup Q ∈ N_{G}(H)/H with H ≠ 1, and restrict C^{(q)}(H) to Q, we obtain a finite dimensional projective RQ-complex (see [6, Lemma 3.6]). This forces Q to be a rank one subgroup. So, to be able to perform the above adding process, we must have the condition that for every p-group 1 ≠ H, the normalizer quotients W_{G}(H) = N_{G}(H)/H have q-rank ≤ 1. Note that a similar argument also shows that for q ∉ 𝒮_{G}, we must have rank_{q} G ≤ 1. This explains the rank conditions that appear in Theorem 5.2 and Theorem 6.1.

We may also assume that \underline{n}(H)+1 is a multiple of the q-period of W_{G}(H) and the gaps between non-zero homology dimensions are large enough: more precisely, for all K, L ∈ 𝒥 with \underline{n}(K) > \underline{n}(L), we have \underline{n}(K) − \underline{n}(L) ≥ l(Γ_{G}), where l(Γ_{G}) denotes the length of the longest chain of maps in the category Γ_{G}. We can easily guarantee both of these conditions by taking joins of the preliminary local models we have constructed. The main result of this section is the following: Theorem 5.2. Let G be a finite group and let R = Z_{(q)}. Suppose that C is an algebraic homotopy representation over R, such that

(i) C an (oriented) R-homology \underline{n}^{(q)}-sphere of projective RΓ_{G}(q)-modules;

(ii) If 1 ≠ H ∈ 𝒥_{p}, then rank_{q}(N_{G}(H)/H) ≤ 1, for every prime p ≠ q.

Then there exists an algebraic homotopy representation C^{(q)} over R, which is an (oriented) R-homology \underline{n}-sphere over RΓ_{G}.

We will add the homology specified by the dimension function \underline{n}^{(p)}, at a prime p ≠ q, by an inductive construction using the number of nonzero homology dimensions. The starting point of the induction is the given complex C. Let n_{1} > n_{2} > ⋯ > n_{s} denote the set of dimensions \underline{n}(H), over all H ∈ 𝒥_{p}. Note that, since the dimension function \underline{n} comes from a unitary representation, we have n_{s} ≥ 1. Let us denote by ℱ_{i}, the collection of subgroups H ∈ 𝒥_{p} such that \underline{n}(H) = n_{i}.

Suppose that we have constructed a finite projective chain complex C over RΓ_{G}, satisfying the conditions (i)-(iii) of Definition 2.6, which has the property that HomDim C(H) = \underline{n}(H) for all H ∈ ℱ_{≤k} where ℱ_{≤k} = ⋃_{i≤k} ℱ_{i}. Our goal is to construct a new finite dimensional projective complex D which also satisfies the conditions (i)-(iii) of Definition 2.6, and has the property that HomDim D(H) = \underline{n}(H) for all H ∈ ℱ_{i} with i ≤ k + 1.

We will construct the complex D as an extension of C by a finite projective chain complex whose homology is isomorphic to the homology that we need to add. Note that since the constructed chain complex D must satisfy the conditions (i)-(iii), the homology we need to add should satisfy the condition that for every H ≤ K with H, K ∈ ℱ_{k+1}, the restriction map on the added homology module is an R-homology isomorphism. Definition 5.3. Let J_{i} denote the RΓ_{G}-module which has the values J_{i}(H) = R for all H ∈ ℱ_{i}, and otherwise J_{i}(H) = 0. The restriction maps r_{H}^{K}: J_{i}(K) → J_{i}(H) for every H, K ∈ ℱ_{i} such that H ≤ K, and the conjugation maps c^{g} : J_{i}(K) → J_{i}(^{g}K) for every K ∈ ℱ and g ∈ G, are assumed to be the identity maps.

In this notation, the chain complex D must have homology isomorphic to J_{i} in dimension n_{i} for all i ≤ k +1, and in dimension zero the homology of D should be isomorphic to \underline{R} restricted to ℱ_{k+1}. It is in general a difficult problem to find chain projective complexes whose homology is given by a block of R-modules with prescribed restriction maps. But in our situation we will be able to do this using some special properties of the poset of subgroups in ℱ_{i} coming from condition (iii) (see Corollary 2.8).

Lemma 5.4. For 1 ≤ i ≤ s, each poset ℱ_{i} is a disjoint union of components where each component has a unique maximal subgroup up to conjugacy.

Note that for every K ∈ 𝒥_{p}, the Sylow q-subgroup of the normalizer quotient W_{G}(K) = N_{G}(K)/K has q-rank equal to one, hence it is q-periodic.

The proof of Theorem 5.2. By our starting assumption, the q-period of W_{G}(K) divides \underline{n}(K) + 1. So by Swan [13], there exists a periodic projective resolution P with

0 → R → P_{n} → ⋯ → P_{1} → P_{0} → R → 0

over the group ring RW_{G}(K) where n = \underline{n}(K). Note that this statement includes the possibility that Sylow q-subgroup of W_{G}(K) is trivial since in that case R would be projective as a RW_{G}(K)-module, and we can easily find a chain complex of the above form by adding a split projective chain complex.

Now suppose that K ∈ 𝒥_{p} is such that (K) is a maximal conjugacy class in ℱ_{k+1}. Consider the RΓ_{G}-complex E_{K}P where E_{K} denotes the extension functor defined in [6, Sect. 2C]. By definition

E_{K}(P)(H) = P ⊗_{R[W_{G}(K)]} R[(G/K)^{H}]

for every H ∈ ℱ. We define the chain complex E_{k+1}P as the direct sum of the chain complexes E_{K}P over all representatives of isomorphism classes of maximal elements in ℱ_{k+1}. Let N denote the subcomplex of E_{k+1}(P) obtained by restricting E_{K}(P) to subgroups H ∈ ℱ_{≤k}. Let I_{k+1}P denote the quotient complex E_{k+1}(P)/N. We have the following:

Lemma 5.5. The homology of I_{k+1}P is isomorphic to J_{k+1} at dimensions 0 and n_{k+1} and zero everywhere else.

Proof. The homology of I_{k+1}P at H ∈ ℱ_{k+1} is isomorphic to

⨁{R ⊗_{R[W_{G}(K)]} R[(G/K)^{H}] : (K) maximal in ℱ_{k+1}}

at dimensions 0 and n_{k+1} and zero everywhere else. Note that (G/K)^{H} = {gK: H^{g} ≤ K}. If gK is such that H^{g} ≤ K, then H ≤ ^{g}K. Now by condition (iii), we must have ⟨K,^{g}K⟩ ∈ ℱ_{k+1}. But (K) was a maximal conjugacy class in ℱ_{k+1}, so we must have K = ^{g}K, hence g ∈ N_{G}(K). This gives 1 ⊗ gK = 1 ⊗ 1 in R ⊗_{R[W_{G}(K)]} R[(G/K)^{H}]. Therefore

R ⊗_{R[W_{G}(K)]} R[(G/K)^{H}] ≅ R

for every H ∈ ℱ_{k+1}. Also, by the same argument H can not be included in two different maximal subgroups in ℱ_{k+1}. So we have I_{k+1}(P)(H) ≅ R for all H ∈ ℱ_{k+1}. Since the restriction maps are given by the inclusion map of fixed point sets (G/H)^{U} ↪ (G/H)^{V} for every U, V ∈ ℱ_{k+1} with V ≤ U, we can conclude that all restriction maps are identity maps. This completes the proof of the lemma. □

The above lemma shows that the homology of I_{k+1}P is exactly the RΓ_{G}-module that we would like to add to the homology of C. To construct D we use an idea similar to the idea used in [6, Section 9B]. Observe that for every RΓ_{G}-chain map f: N → C, there is a push-out diagram of chain complexes

[figure: 0; N; f; E_{K}P; I_{k+1}P; 0; 0; C; C_{f}; I_{k+1}P; 0 .]

The homology of N is only nonzero at dimensions 0 and n_{k+1} and at these dimensions the homology is only nonzero at subgroups H ∈ ℱ_{≤k}. At these subgroups the homology of N(H) is isomorphic to the direct sum of modules of the form R ⊗_{RW_{G}(K)} R[(G/K)^{H}]. Note that for every H ∈ ℱ_{≤k}, there is an augmentation map

ε_{H}: R ⊗_{RW_{G}(K)} R[(G/K)^{H}] → R

which takes r ⊗ gK to r for every r ∈ R. The collection of these maps gives a map of RΓ_{G}-modules denoted ε_{K}: E_{K}R → R. Taking the sum over all conjugacy classes of maximal subgroups, we get a map ∑_{K} ε_{K}: ⊕_{K} E_{K}R → R. Repeating the arguments given in [6, Section 9B], it is easy to see that if f is a chain map such that the induced map on zeroth homology f_{∗}: H_{0}(N) → H_{0}(C) is the same map as the sum of augmentation maps ∑_{K} ε_{K}, then the chain complex C_{f} will have the identity map as the restriction maps on zeroth homology. At dimension n_{k+1} we will have zero map since the homology of C is zero at dimension n_{k+1} by assumption.

Unfortunately, we can not take D as C_{f} since the complex I_{k+1}P is not projective in general, and neither is N. We note that finding a chain map N satisfying the given condition is not a easy task without projectivity (compare [6, Section 9B], where this complex was a projective). So we first need to replace the sequence 0 → N → E_{k+1}P → I_{k+1}P → 0 with a sequence of projective chain complexes.

Lemma 5.6. There is a diagram of chain complexes where all the complexes P′, P′′, P′′′ are finite projective chain complexes over RΓ_{G} and all the vertical maps induce isomorphisms on homology:

\xymatrix{0 \ar[r] & P′ \ar[d] \ar[r] & P′′ \ar[d] \ar[r] & P′′′ \ar[d] \ar[r] & 0 \\ 0 \ar[r] & N \ar[r] & E_{k+1}P \ar[r] & I_{k+1}P \ar[r] & 0 .}

Proof. Since E_{K}P is a projective chain complex of length n, E_{k+1}P is a finite projective chain complex. So, by [11, Lemma 11.6], it is enough to show that N is weakly equivalent to a finite projective complex. For this first note that N = ⨁ N_{K} is a direct sum of chain complexes N_{K} where N_{K} is the restriction of E_{K}P to subgroups H ∈ ℱ_{≤k}. So it is enough to show that N_{K} is weakly equivalent to a finite projective chain complex. To prove this, we will show that for each i, the RΓ_{G}-module N_{i} := (N_{K})_{i} has a finite projective resolution. The module N_{i} is nonzero only at subgroups H ∈ ℱ_{≤k} and at each such a subgroup, we have

N_{i}(H) = (E_{K}P_{i})(H) = P_{i} ⊗_{RW_{G}(K)} R[(G/K)^{H}].

So, as an RW_{G}(H)-module N_{i}(H) is a direct summand of R[(G/K)^{H}] which is isomorphic to

⨁{R[W_{G}(H)/W^{g}_{K}(H)] : K-conjugacy classes of subgroups H^{g} ≤ K}

as an RW_{G}(H)-module. Since K is a p-group, these modules are projective over the ground ring R because R is q-local. So, for each H ∈ ℱ_{≤k}, the RW_{G}(H)-module N_{i}(H) is projective. Now consider the map

π: ⊕_{H} E_{H}N_{i}(H) → N_{i}

induced by maps adjoint to the identity maps at each H. We can take ⊕_{H}E_{H}N_{i}(H) as the first projective module of the resolution, and consider the kernel Z_{0} of π: ⊕_{H} E_{H}N_{i}(H) → N_{i}. Note that Z_{0} has smaller length and it also have the property that at each L, the W_{G}(L) modules Z_{0}(L) are projective. This follows from the fact that R[(G/H)^{L}] is projective as a W_{G}(L)-module by the same argument we used above. Continuing this way, we can find a finite projective resolution for N_{i} of length ≤ l(Γ). □

Now it remains to show that there is a chain map f: P′ → C such that the induced map on zeroth homology f_{∗}: H_{0}(P′) ≅ H_{0}(N) → H_{0}(C) is given by the sum of augmentation maps ε_{K} over the conjugacy classes of maximal subgroups K in ℱ_{k+1}. Then the complex D will be defined as the push-out complex that fits into the diagram

[figure: 0; P′; f; P′′; P′′′; 0; 0; C; D; P′′′; 0 .]

Since both C and P′′′ are finite projective chain complexes, D will also be a finite projective complex.

To construct the chain map f: P′ → C, first note that the chain complex C has no homology below dimension n_{k}. By assumption on the gaps between nonzero homology dimensions, we can assume that n_{k} ≥ n_{k+1} + l(Γ_{G}) ≥ l(P′). So, starting with the sum of augmentation maps ∑_{K} ε_{K} at dimensions zero, we can obtain a chain map as follows:

[figure: 0; P′_{m}; f_{m}; ∂_{m}^{P′}; ⋯; P′_{0}; f_{0}; H_{0}(N); ∑_{K} ε_{K}; 0; C_{m+1}; C_{m}; ∂_{m}^{C}; ⋯; C_{0}; H_{0}(C); 0]

where m = l(P′). This completes the proof of Theorem 5.2. □

6. The Proof of Theorem A

In this section we establish our main technique for constructing actions on homotopy spheres, based on a given 𝒫-representation, where 𝒫 = ⋃{ℱ_{p} | p ∈ 𝒮_{G}} denote the family of all p -subgroups of G, for the primes p in a given set 𝒮_{G} (see Definitions 4.1 and 4.4). Theorem A stated in the introduction will follow from this theorem almost immediately once we use the family of p-effective characters constructed by M. A. Jackson [9]. The main technical theorem is the following:

Theorem 6.1. Let G be a finite group and let 𝒮_{G} = {p | rank_{p} G ≥ 2}. Suppose that (i) V(ℱ_{p}) is a ℱ_{p}-representation for G, with Iso(V(ℱ_{p})) = 𝒥_{p}, for each p ∈ 𝒮_{G}; (ii) If p ∈ 𝒮_{G} and 1 ≠ H ∈ 𝒥_{p}, then we have rank_{q}(N_{G}(H)/H) ≤ 1 for every q ≠ p.

Then there exists a finite G-CW-complex X ≃ S^{n}, with isotropy in 𝒥 = ⋃{𝒥_{p} | p ∈ 𝒮_{G}}, which is a geometric homotopy representation for G.

Remark 6.2. The construction we give in the proof of Theorem 6.1 gives a simplyconnected homotopy representation X, with dim X^{H} ≥ 3, for all H ∈ 𝒥, whenever X^{H} ≠ ∅. It also relates the dimension function of X to the linear dimension functions Dim S(V_{H}), for V_{H} ∈ ⋃{V(ℱ_{p}) | p ∈ 𝒮_{G}} in the following way: for every prime p ∈ 𝒮_{G}, there exists an integer k_{p} > 0 such that for every H ∈ ℱ_{p}, the equality dim X^{H} = dim S(V_{H}^{⊕k_{p}})^{H} holds.

As we discussed in the previous section, the condition on the q-rank of N_{G}(H)/H is a necessary condition for the existence of such actions. Recall that this condition is used in an essential way in the proof of Theorem 5.2.

The proof of Theorem 6.1. By the realization theorem (Theorem 3.1) proved in Section 3, we only need to construct a chain complex of ZΓ_{G}-modules satisfying the conditions (i), (ii) and (iii) of Definition 2.6. If we apply Theorem 5.2 to the preliminary local model constructed in Section 4, we obtain a finite projective complex C^{(p)}, over the orbit category Z_{(p)}Γ_{G} with respect to the family 𝒥, for each prime p dividing the order of G. In addition, C^{(p)} is an oriented Z_{(p)}-homology \underline{n}-sphere, with the same dimension function \underline{n} = HomDim C^{(p)}(0) coming from the preliminary local models. By construction, the complex C^{(p)} satisfies the conditions (i), (ii) and (iii) of Definition 2.6 for R = Z_{(p)}.

We may also assume that \underline{n}(H) ≥ 3 for every H ∈ 𝒥, and that the gaps between non-zero homology dimensions have the following property: for all K, L ∈ 𝒥 with \underline{n}(K) > \underline{n}(L), we have \underline{n}(K) − \underline{n}(L) ≥ l(Γ_{G}) where l(Γ_{G}) denotes the length of the longest chain of maps in the category Γ_{G}.

To complete the proof of Theorem 6.1, we first need to glue these complexes C^{(p)} together to obtain an algebraic \underline{n}-sphere over ZΓ_{G}. By [6, Theorem 6.7], there exists a finite projective chain complex C of ZΓ_{G}-modules, which is a Z-homology \underline{n}-sphere, such that Z_{(p)} ⊗ C is chain homotopy equivalent to the local model C^{(p)}, for each prime p dividing the order of G. The complex C has a (possibly non-zero) finiteness obstruction (see Lueck [11, §10-11]), but this can be eliminated by joins (see [6, §7]).

After applying [6, Theorem 7.6], we may assume that C is a finite free chain complex of ZΓ_{G}-modules which is a Z-homology \underline{n}-sphere. Moreover, C is an algebraic homotopy representation: it satisfies the conditions (i), (ii) and (iii) of Definition 2.6 for R = Z, since these conditions hold locally at each prime.

We have now established all the requirements for Theorem 3.2. For the family ℱ used in its statement, we use ℱ = 𝒥. For all H ∈ ℱ, we have the condition \underline{n}(H) ≥ 3. Now Theorem 3.2 gives a finite G-CW-complex X ≃ S^{n} with isotropy 𝒥 such that X^{H} is an homotopy sphere for every H ∈ 𝒥. □

Now we are ready to prove Theorem A.

The proof of Theorem A. Let G be a rank 2 finite group and let 𝒮_{G} denote the set of primes with rank_{p} G = 2. Since it is assumed that G does not p′-involve Qd(p) for any odd prime p, we can apply [9, Theorem 47] and obtain a p-effective representation V_{p}, for every prime p ∈ 𝒮_{G}. We apply Theorem 6.1 to the ℱ_{p}-representations V(ℱ_{p}) given by this collection {V_{p}} (see Example 4.3). Since V_{p} is p-effective means that all isotropy subgroups in ℋ_{p} are rank one p-subgroups (see Example 4.11), the isotropy is contained in the family ℋ of rank one p-subgroups of G, for all p ∈ 𝒮_{G}. We therefore obtain a finite G-CW-complex X ≃ S^{n}, with rank 1 isotropy in ℋ, such that X^{H} is an homotopy sphere (possibly empty) for every H ∈ ℋ. □

The proof of Corollary B follows easily from Theorem A since if rank_{q}(G) ≤ 1, then for every p-group H, we must have rank_{q}(N_{G}(H)/H) ≤ 1. So we can apply Theorem A to obtain Corollary B.

Note that the condition about Qd(p) being not p′-involved in G is a necessary condition for the existence of actions of rank 2 groups on finite CW-complexes X ≃ S^{n} with rank one isotropy.

Proposition 6.3. If G acts on a finite complex X homotopy equivalent to a sphere with rank one isotropy, then G cannot p′-involve Qd(p) for any odd prime p.

Proof. Suppose that G has a normal p′-subgroup K such that Qd(p) is isomorphic to a subgroup in N_{G}(K)/K. Let L be subgroup of G such that K ◁ L ≤ N_{G}(K) and L/K ≅ Qd(p). The quotient group Q = L/K acts on the orbit space Y = X/K via the action defined by (gK)(Kx) = Kgx for every g ∈ L and x ∈ X.

We observe two things about this action. First, by a transfer argument [1, Theorem 2.4, p. 120], the space Y has the mod p homology of a sphere. Second, all the isotropy subgroups of the Q-action on Y have p-rank ≤ 1. To see this, let Q_{y} denote the isotropy subgroup at y ∈ Y and let x ∈ X be such that y = Kx. It is easy to see that Q_{y} = L_{x}K/K ≅ L_{x}/(L_{x} ∩ K). Since K is a p′-group, this shows that p-subgroups of Q_{y} are isomorphic to p-subgroups of the isotropy subgroup L_{x}. Since L acts on X with rank one isotropy, we conclude that rank_{p}(Q_{y}) ≤ 1 for every y ∈ Y.

Now the rest of the proof follows from the argument given in Ünlü [16, Theorem 3.3]. Let P be a Sylow p-subgroup of Q ≅ Qd(p). Then P is an extra-special p-group of order p^{3} with exponent p (since p is odd). Let c denote a central element and a a noncentral element in P. Since the P -action on Y has rank 1 isotropy subgroups, we have Y^{E} = ∅ for every rank two p-subgroup E ≤ P. Therefore Y^{⟨c⟩} = ∅ by Smith theory, since otherwise P/⟨c⟩ ≅ Z/p × Z/p would act freely on Y^{⟨c⟩} which is a mod p homology sphere.

Now consider the subgroup E = ⟨a, c⟩. Since ⟨a⟩ and ⟨c⟩ are conjugate in Q, all cyclic subgroups of E are conjugate. In particular, we have Y^{H} = ∅ for every cyclic subgroup H in E. This is a contradiction, since E cannot act freely on Y. □

7. Discussion and examples

We first discuss the rank conditions in the statement of Theorem A. Suppose that X is a finite G-CW-complex. Recall that Iso(X) = {H | H ≤ G_{x} for some x ∈ X} denotes the minimal family containing all the isotropy subgroups of the G-action on X. We call this the isotropy family. Note that H ∈ Iso(X) if and only if X^{H} ≠ ∅. We say that X has rank k isotropy if rank G_{x} ≤ k for all x ∈ X and there exists a subgroup H with rank H = k and X^{H} ≠ ∅. Let 𝒫 denote the family of all prime-power order subgroups of G.

Lemma 7.1. Let G be a finite group, and let X be a finite G-CW-complex with X ≃ S^{n}. (i) If H is a maximal p-subgroup in Iso(X), then rank_{p}(N_{G}(H)/H) ≤ 1. (ii) If 1 ≠ H ∈ Iso(X) ⊆ 𝒫 is a p-subgroup, and X^{H} is an integral homology sphere,

then rank_{q}(N_{G}(H)/H) ≤ 1, for all primes q ≠ p.

Proof. This follows from two basic results of P. A. Smith theory [1, III.8.1]), which state (i) that the fixed set of a p-group action on a finite-dimensional mod p homology sphere is again a mod p homology sphere (or the empty set), and (ii) that Z/p × Z/p can not act freely on a finite G-CW-complex X with the mod p homology of a sphere.

For any prime p dividing the order of G, let H ∈ Iso(X) denote a maximal p-subgroup with X^{H} ≠ ∅. For any x ∈ X^{H}, we have H ≤ G_{x} and if g · x = x, for some g ∈ N_{G}(H) of p-power order, it follows that the subgroup ⟨H, g⟩ ≤ G_{x}. Since H was a maximal p-subgroup in Iso(X), we conclude that g ∈ H. Therefore N_{G}(H)/H acts freely on the fixed set X^{H}, which is a mod p homology sphere, and hence rank_{p}(N_{G}(H)/H) ≤ 1.

If q ≠ p and H is a p-subgroup in Iso(X) ⊆ 𝒫, then any q-subgroup Q of N_{G}(H)/H must act freely on X^{H} (since x ∈ X^{H} implies G_{x} is a p-group). Since X^{H} is assumed to be an integral homology sphere, Smith theory implies that rank_{q}(Q) ≤ 1. □

Example 7.2. If G is the extra-special p-group of order p^{3}, then the centre Z(G) = Z/p can not be a maximal isotropy subgroup in Iso(X). On the other hand, we know that G acts on a finite complex X ≃ S^{n} with rank one isotropy: just take the linear sphere S(Ind_{Z(G)}^{G} W) for some nontrivial one-dimensional representation W of Z(G). So we can not require that G acts on X ≃ S^{n} with Iso(X) containing all rank one subgroups.

For any prime p, we can restrict the G-action on X to a p-subgroup of maximal rank.

Corollary 7.3. If X is a finite G-CW-complex with X ≃ S^{n} and rank k isotropy, then rank_{p} G ≤ k + 1, for all primes p.

These results help to explain the rank conditions in Theorem A. If we have rank one isotropy, then we must assume that G has rank two. However, condition (ii) on the q-rank of the normalizer quotient is only necessary for p-subgroups H, with q ≠ p, for which X^{H} ≠ ∅ is an integral homology sphere. In order to get a complete list of necessary conditions, we must have more precise control of the structure of the isotropy subgroups.

Now we discuss two applications of Theorem A and Theorem 6.1.

Example 7.4. The alternating group G = A_{6} admits a finite G-CW-complex X ≃ S^{n}, with rank one isotropy. This follows from Theorem A once we verify that G satisfies the necessary conditions. Note that A_{6} has order 2^{3} · 3^{2} · 5 = 360 so it automatically satisfies the condition about Qd(p), since it can not include an extra-special p-group of order p^{3} for an odd prime p. For the q-rank condition, note that 𝒮_{G} = {2, 3}, so we need to check this condition only for primes p = 2 and 3. Here are some easily verified facts:

• A 2-Sylow subgroup P ≤ G is isomorphic to the dihedral group D_{8}, so all rank one 2-subgroups are cyclic, and ℋ_{2} = {1, C_{2}, C_{4}}.

• N_{G}(C_{2}) = P, and rank_{3}(N_{G}(C_{2})/C_{2}) = 0.

• N_{G}(C_{4}) = P and rank_{3}(N_{G}(C_{4})/C_{4}) = 0. Now, let Q be a Sylow 3-subgroup in G. Then Q ≅ C_{3} × C_{3}.

• Any subgroup of order 3 in G is conjugate to C_{3}^{A} = ⟨(123)⟩ or C_{3}^{B} = ⟨(123)(456)⟩.

• |N_{G}(C_{3}^{A})/C_{3}^{A}| = 6 and rank_{2}(N_{G}(C_{3}^{A})/C_{3}^{A}) = 1.

• |N_{G}(C_{3}^{B})/C_{3}^{B}| = 6 and rank_{2}(N_{G}(C_{3}^{B})/C_{3}^{B}) = 1 We conclude that condition (ii) of Theorem A holds for this group.

We now give an example which does not satisfy the q-rank conditions in Theorem A, but where we can apply Theorem 6.1 directly.

Example 7.5. The alternating group G = A_{7} admits a finite G-CW-complex X ≃ S^{n}, with rank one isotropy. The order of G is 2^{3} · 3^{2} · 5 · 7, so this group also automatically satisfies the Qd(p) condition. Here is a summary of the main structural facts:

• The 3-Sylow subgroup Q ≤ G is isomorphic to C_{3} × C_{3}.

• Any subgroup of order 3 in G is conjugate to C_{3}^{A} = ⟨(123)⟩ or C_{3}^{B} = ⟨(123)(456)⟩.

• The Sylow 2-subgroup of N_{G}(C_{3}^{A}) is isomorphic to D_{8}.

• |N_{G}(C_{3}^{A})/C_{3}^{A}| = 24 and rank_{2}(N_{G}(C_{3}^{A})/C_{3}^{A}) = 2.

• N_{G}(C_{3}^{B}) ≅ (C_{3} × C_{3}) ⋊ C_{2} and rank_{2}(N_{G}(C_{3}^{B})/C_{3}^{B}) = 1.

• |N_{G}(C_{2})| = 24, and rank_{3}(N_{G}(C_{2})/C_{2}) = 1.

• N_{G}(C_{4}) ≅ D_{8} and rank_{3}(N_{G}(C_{4})/C_{4}) = 0 We see that the rank condition in Theorem A fails for 3-subgroups, since there is a cyclic 3subgroup H = C_{3}^{A} with rank_{2}(N_{G}(H)/H) = 2. On the other hand, by applying Theorem 6.1 directly, we can still find a finite G-CW-complex X ≃ S^{n}, with rank one isotropy in the family generated by {1, C_{2}, C_{4}, C_{3}^{B}}.

In this case, we have 𝒮_{G} = {2, 3}. For p = 2, we can use the ℱ_{2}-representation V_{2} from [9], since A_{7} satisfies the rank condition for 2-subgroups. It remains to show that there exists an ℱ_{3}-representation of G with isotropy subgroups only type B cyclic 3-subgroups. But this is easily constructed by taking V_{3} as the direct sum of augmented permutation modules I(Q/K_{1}) ⊕ I(Q/K_{2}) where K_{1} = ⟨(123)(456)⟩ and K_{2} = ⟨(123)(465)⟩. It is clear that this representation respects fusion, and has isotropy given only by the cyclic 3-subgroups of type B lying in ℱ_{3}.

Remark 7.6. When G is a finite group with a rank two elementary abelian Sylow qsubgroup Q, the representation

V_{q} = ⨁{I(Q/K_{i}) : 1 ≤ i ≤ s}

over some family of rank 1 subgroups K_{i}, which is closed under G-conjugacy, will give a q-effective representation which respects fusion. But for more general Sylow q-subgroups, the above representation may fail to be q-effective. Note that for V_{q} to be q-effective one needs to have exactly one double coset in E\Q/K_{i} for every K_{i} and for every rank 2 elementary abelian subgroup E of Q. This fails, for example, if Q = C_{p^{2}} × C_{p^{2}} is a non-elementary rank two abelian group.

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Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada

E-mail address: hambleton@mcmaster.ca

Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey

E-mail address: yalcine@fen.bilkent.edu.tr